Composite functions and intermediate domains

One machine’s output becomes another machine’s input. An allowed starting input can still produce an invalid intermediate value.

• One machine’s output becomes another machine’s input. An allowed starting input can still produce an invalid intermediate value.
• This lesson studies composition 复合函数: Applying one function to the output of another, in a specified order.

Choose the mathematical structure

• Here fg means f composed with g: fg(x)=f(g(x)), not multiplication. Apply g first. A composite input must belong to the inner function’s domain, and its output must belong to the outer function’s domain. In general fg and gf have different formulas and different domains.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

$$(f\circ g)(x)=\sqrt{\frac1{x-1}-2},\qquad1

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Example

Let f(x)=√(x−2), with x≥2, and g(x)=1/(x−1), with x≠1. Then fg(x)=√(1/(x−1)−2). It requires 1/(x−1)≥2. A negative denominator cannot meet this positive lower bound; for x>1, multiplying by x−1 preserves order and gives x≤3/2. Thus the domain is 1<x≤3/2. In the reverse order, gf(x)=1/(√(x−2)−1). It requires x≥2 and √(x−2)≠1, so x≠3. Its domain is [2,3)∪(3,∞). At x=6, gf(6)=1, while fg(6) is not real.

Test a tempting shortcut

• Checking only the inner domain is insufficient. Do not multiply the inequality by x−1 before deciding its sign. An excluded intermediate denominator remains excluded even if another form appears simpler. A formula and domain together define the composite function.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Trace the intermediate value: gf(6) uses f(6)=2 then g(2)=1. For fg(5/4), g(5/4)=4 then f(4)=√2. At x=3, f(3)=1 is valid, but g(1) is undefined, so gf(3) is undefined. Mark every retained or excluded endpoint in interval notation.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• 7357 · A-level · B. Match the target tier and specification before assigning extensions.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.